Related Formula
μ = (Σ fᵢ xᵢ)/(Σ fᵢ)$$\mu = \frac{\sum f_i x_i}{\sum f_i}$$
σ² = (Σ fᵢ xᵢ²)/(Σ fᵢ) - μ²$$\sigma^2 = \frac{\sum f_i x_i^2}{\sum f_i} - \mu^2$$
Core Logic
Midpoints (xᵢ$x_i$): 6, 10, 14, 18$6, 10, 14, 18$.
Frequencies (fᵢ$f_i$): 3, λ, 4, 7$3, \lambda, 4, 7$.
Sum of frequencies Σ fᵢ = 3 + λ + 4 + 7 = 14 + λ$\sum f_i = 3 + \lambda + 4 + 7 = 14 + \lambda$.
Calculate Mean (μ$\mu$):
μ = (6(3) + 10(λ) + 14(4) + 18(7))/(14 + λ)$$\mu = \frac{6(3) + 10(\lambda) + 14(4) + 18(7)}{14 + \lambda}$$
μ = (18 + 10λ + 56 + 126)/(14 + λ) = (10λ + 200)/(λ + 14) = 10 + (60)/(λ + 14)$$\mu = \frac{18 + 10\lambda + 56 + 126}{14 + \lambda} = \frac{10\lambda + 200}{\lambda + 14} = 10 + \frac{60}{\lambda + 14}$$
Since data is standard, μ$\mu$ and λ$\lambda$ typically hold integer values, hinting λ + 14$\lambda + 14$ must be a factor of 60. Therefore λ$\lambda$ could be 1, 6, 16$1, 6, 16$.
Step 1: Applying Variance
Variance σ² = 19$\sigma^2 = 19$.
σ² = (Σ fᵢ xᵢ²)/(14 + λ) - μ²$$\sigma^2 = \frac{\sum f_i x_i^2}{14 + \lambda} - \mu^2$$
Σ fᵢ xᵢ² = 3(36) + λ(100) + 4(196) + 7(324) = 108 + 100λ + 784 + 2268 = 100λ + 3160$$\sum f_i x_i^2 = 3(36) + \lambda(100) + 4(196) + 7(324) = 108 + 100\lambda + 784 + 2268 = 100\lambda + 3160$$
Substitute into variance:
19 = (100λ + 3160)/(λ + 14) - ((10λ + 200)/(λ + 14))²$$19 = \frac{100\lambda + 3160}{\lambda + 14} - \left(\frac{10\lambda + 200}{\lambda + 14}\right)^2$$
Step 2: Testing Candidates
Let's test the potential integer λ = 6$\lambda = 6$ (a factor of 60 is 14+6=20$14+6=20$):
If λ = 6$\lambda = 6$:
μ = 10 + (60)/(20) = 10 + 3 = 13$$\mu = 10 + \frac{60}{20} = 10 + 3 = 13$$
Let's verify variance for λ = 6$\lambda = 6$:
σ² = (100(6) + 3160)/(20) - 13² = (3760)/(20) - 169 = 188 - 169 = 19$$\sigma^2 = \frac{100(6) + 3160}{20} - 13^2 = \frac{3760}{20} - 169 = 188 - 169 = 19$$
This perfectly matches the given variance.
Thus, λ = 6$\lambda = 6$ and μ = 13$\mu = 13$.
λ + μ = 6 + 13 = 19$$\lambda + \mu = 6 + 13 = 19$$
Pattern Recognition
Instead of solving a horrible cubic/quadratic equation for the variance parameter, use the integer division trait of the mean to test valid candidates.
Chapter Mix
Class 11 Maths: Statistics
